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A Two-Grid Algorithm for Solution of the Difference Equations of a System of Singularly Perturbed Semilinear Equations

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 5434)

Abstract

We propose a two-grid algorithm for implementation of a generalized A.M.Il’in’s scheme to a system of semilinear diffusion convection-dominated equations. To solve the nonlinear algebraic system of difference equations we use Newton method. We derive the difference scheme on a coarse mesh and, then using uniform interpolation, taking into account the boundary layers, we obtain the initial guess for an iterative method on a fine mesh. Estimates of the accuracy and the computational work are obtained. The main advantage of the proposed algorithm is the computational cost.

Keywords

  • Boundary Layer
  • Initial Guess
  • Newton Method
  • Coarse Grid
  • Boundary Layer Problem

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References

  1. Angelova, I.T., Vulkov, L.G.: Comparison of the two-grid method for singularly perturbed reaction-diffusion problems on different meshes. Amer. Inst. of Phys. CP (in press)

    Google Scholar 

  2. Axelsson, O.: On mesh independence and Newton methods. Appl. of Mathematics 4-5(38), 249–265 (1993)

    MATH  Google Scholar 

  3. Il’in, A.M.: A difference scheme for a differential eqution with a small parameter affecting the highest derivative. Mat. Zametki. 6, 237–248 (1969) (in Russian)

    MathSciNet  Google Scholar 

  4. Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problems without turning pointd. Math. Comput. 32(144), 1025–1039 (1978)

    CrossRef  MATH  Google Scholar 

  5. Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow problems. Springer, Berlin (2008)

    MATH  Google Scholar 

  6. Vulkov, L.G., Zadorin, A.I.: Two-grid interpolation algorithms for difference schemes of exponential type for semilinear diffusion convection-dominated equations. Amer. Inst. of Phys. CP (in press)

    Google Scholar 

  7. Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15(1), 231–237 (1994)

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Zadorin, A.I.: Numerical solution of a boundary value problem for a set of equations with a small parameter. Comp. Math. and Math. Phys. 38(8), 1255–1265 (1998)

    MATH  Google Scholar 

  9. Zadorin, A.I.: Method of interpolation for a boundary layer problem. Sib. J. of Numer. Math. 10(3), 267–275 (2007)

    MATH  Google Scholar 

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Vulkov, L.G., Zadorin, A.I. (2009). A Two-Grid Algorithm for Solution of the Difference Equations of a System of Singularly Perturbed Semilinear Equations. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2008. Lecture Notes in Computer Science, vol 5434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00464-3_68

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  • DOI: https://doi.org/10.1007/978-3-642-00464-3_68

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00463-6

  • Online ISBN: 978-3-642-00464-3

  • eBook Packages: Computer ScienceComputer Science (R0)